3.1603 \(\int (b+2 c x) \sqrt{d+e x} (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=252 \[ \frac{8 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^6}+\frac{4 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac{10 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^6}+\frac{4 c^3 (d+e x)^{13/2}}{13 e^6} \]

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2))/(3*e^6) + (4*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 +
b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(5/2))/(5*e^6) - (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d
- 3*a*e))*(d + e*x)^(7/2))/(7*e^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(9/2))/(9*e^6)
- (10*c^2*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^6) + (4*c^3*(d + e*x)^(13/2))/(13*e^6)

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Rubi [A]  time = 0.124536, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {771} \[ \frac{8 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^6}+\frac{4 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac{10 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^6}+\frac{4 c^3 (d+e x)^{13/2}}{13 e^6} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*Sqrt[d + e*x]*(a + b*x + c*x^2)^2,x]

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2))/(3*e^6) + (4*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 +
b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(5/2))/(5*e^6) - (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d
- 3*a*e))*(d + e*x)^(7/2))/(7*e^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(9/2))/(9*e^6)
- (10*c^2*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^6) + (4*c^3*(d + e*x)^(13/2))/(13*e^6)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (b+2 c x) \sqrt{d+e x} \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}{e^5}+\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{3/2}}{e^5}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{5/2}}{e^5}+\frac{4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{7/2}}{e^5}-\frac{5 c^2 (2 c d-b e) (d+e x)^{9/2}}{e^5}+\frac{2 c^3 (d+e x)^{11/2}}{e^5}\right ) \, dx\\ &=-\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{3 e^6}+\frac{4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{5 e^6}-\frac{2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^6}+\frac{8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{9 e^6}-\frac{10 c^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^6}+\frac{4 c^3 (d+e x)^{13/2}}{13 e^6}\\ \end{align*}

Mathematica [A]  time = 0.380214, size = 291, normalized size = 1.15 \[ \frac{2 (d+e x)^{3/2} \left (-286 c e^2 \left (21 a^2 e^2 (2 d-3 e x)-9 a b e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^2 \left (-48 d^2 e x+32 d^3+60 d e^2 x^2-70 e^3 x^3\right )\right )+429 b e^3 \left (35 a^2 e^2+14 a b e (3 e x-2 d)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+13 c^2 e \left (44 a e \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )+5 b \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )\right )-10 c^3 \left (480 d^3 e^2 x^2-560 d^2 e^3 x^3-384 d^4 e x+256 d^5+630 d e^4 x^4-693 e^5 x^5\right )\right )}{45045 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*Sqrt[d + e*x]*(a + b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(3/2)*(-10*c^3*(256*d^5 - 384*d^4*e*x + 480*d^3*e^2*x^2 - 560*d^2*e^3*x^3 + 630*d*e^4*x^4 - 693*e
^5*x^5) + 429*b*e^3*(35*a^2*e^2 + 14*a*b*e*(-2*d + 3*e*x) + b^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 286*c*e^2*(
21*a^2*e^2*(2*d - 3*e*x) - 9*a*b*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + b^2*(32*d^3 - 48*d^2*e*x + 60*d*e^2*x^2 -
70*e^3*x^3)) + 13*c^2*e*(44*a*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 5*b*(128*d^4 - 192*d^3*e
*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4))))/(45045*e^6)

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Maple [A]  time = 0.007, size = 359, normalized size = 1.4 \begin{align*}{\frac{13860\,{c}^{3}{x}^{5}{e}^{5}+40950\,b{c}^{2}{e}^{5}{x}^{4}-12600\,{c}^{3}d{e}^{4}{x}^{4}+40040\,a{c}^{2}{e}^{5}{x}^{3}+40040\,{b}^{2}c{e}^{5}{x}^{3}-36400\,b{c}^{2}d{e}^{4}{x}^{3}+11200\,{c}^{3}{d}^{2}{e}^{3}{x}^{3}+77220\,abc{e}^{5}{x}^{2}-34320\,a{c}^{2}d{e}^{4}{x}^{2}+12870\,{b}^{3}{e}^{5}{x}^{2}-34320\,{b}^{2}cd{e}^{4}{x}^{2}+31200\,b{c}^{2}{d}^{2}{e}^{3}{x}^{2}-9600\,{c}^{3}{d}^{3}{e}^{2}{x}^{2}+36036\,{a}^{2}c{e}^{5}x+36036\,a{b}^{2}{e}^{5}x-61776\,abcd{e}^{4}x+27456\,a{c}^{2}{d}^{2}{e}^{3}x-10296\,{b}^{3}d{e}^{4}x+27456\,{b}^{2}c{d}^{2}{e}^{3}x-24960\,b{c}^{2}{d}^{3}{e}^{2}x+7680\,{c}^{3}{d}^{4}ex+30030\,b{a}^{2}{e}^{5}-24024\,{a}^{2}cd{e}^{4}-24024\,a{b}^{2}d{e}^{4}+41184\,abc{d}^{2}{e}^{3}-18304\,a{c}^{2}{d}^{3}{e}^{2}+6864\,{b}^{3}{d}^{2}{e}^{3}-18304\,{b}^{2}c{d}^{3}{e}^{2}+16640\,b{c}^{2}{d}^{4}e-5120\,{c}^{3}{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)^2*(e*x+d)^(1/2),x)

[Out]

2/45045*(e*x+d)^(3/2)*(6930*c^3*e^5*x^5+20475*b*c^2*e^5*x^4-6300*c^3*d*e^4*x^4+20020*a*c^2*e^5*x^3+20020*b^2*c
*e^5*x^3-18200*b*c^2*d*e^4*x^3+5600*c^3*d^2*e^3*x^3+38610*a*b*c*e^5*x^2-17160*a*c^2*d*e^4*x^2+6435*b^3*e^5*x^2
-17160*b^2*c*d*e^4*x^2+15600*b*c^2*d^2*e^3*x^2-4800*c^3*d^3*e^2*x^2+18018*a^2*c*e^5*x+18018*a*b^2*e^5*x-30888*
a*b*c*d*e^4*x+13728*a*c^2*d^2*e^3*x-5148*b^3*d*e^4*x+13728*b^2*c*d^2*e^3*x-12480*b*c^2*d^3*e^2*x+3840*c^3*d^4*
e*x+15015*a^2*b*e^5-12012*a^2*c*d*e^4-12012*a*b^2*d*e^4+20592*a*b*c*d^2*e^3-9152*a*c^2*d^3*e^2+3432*b^3*d^2*e^
3-9152*b^2*c*d^3*e^2+8320*b*c^2*d^4*e-2560*c^3*d^5)/e^6

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Maxima [A]  time = 0.999328, size = 416, normalized size = 1.65 \begin{align*} \frac{2 \,{\left (6930 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{3} - 20475 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 20020 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 6435 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 18018 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 15015 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{45045 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^2*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(6930*(e*x + d)^(13/2)*c^3 - 20475*(2*c^3*d - b*c^2*e)*(e*x + d)^(11/2) + 20020*(5*c^3*d^2 - 5*b*c^2*d
*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(9/2) - 6435*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - (b^
3 + 6*a*b*c)*e^3)*(e*x + d)^(7/2) + 18018*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e^2 - (b^3 + 6*a
*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*(e*x + d)^(5/2) - 15015*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c +
a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d)^(3/2))/e^6

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Fricas [A]  time = 1.42809, size = 927, normalized size = 3.68 \begin{align*} \frac{2 \,{\left (6930 \, c^{3} e^{6} x^{6} - 2560 \, c^{3} d^{6} + 8320 \, b c^{2} d^{5} e + 15015 \, a^{2} b d e^{5} - 9152 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 3432 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 12012 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 315 \,{\left (2 \, c^{3} d e^{5} + 65 \, b c^{2} e^{6}\right )} x^{5} - 35 \,{\left (20 \, c^{3} d^{2} e^{4} - 65 \, b c^{2} d e^{5} - 572 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 5 \,{\left (160 \, c^{3} d^{3} e^{3} - 520 \, b c^{2} d^{2} e^{4} + 572 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} + 1287 \,{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 3 \,{\left (320 \, c^{3} d^{4} e^{2} - 1040 \, b c^{2} d^{3} e^{3} + 1144 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 429 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5} - 6006 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} +{\left (1280 \, c^{3} d^{5} e - 4160 \, b c^{2} d^{4} e^{2} + 15015 \, a^{2} b e^{6} + 4576 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 1716 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 6006 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^2*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(6930*c^3*e^6*x^6 - 2560*c^3*d^6 + 8320*b*c^2*d^5*e + 15015*a^2*b*d*e^5 - 9152*(b^2*c + a*c^2)*d^4*e^2
+ 3432*(b^3 + 6*a*b*c)*d^3*e^3 - 12012*(a*b^2 + a^2*c)*d^2*e^4 + 315*(2*c^3*d*e^5 + 65*b*c^2*e^6)*x^5 - 35*(2
0*c^3*d^2*e^4 - 65*b*c^2*d*e^5 - 572*(b^2*c + a*c^2)*e^6)*x^4 + 5*(160*c^3*d^3*e^3 - 520*b*c^2*d^2*e^4 + 572*(
b^2*c + a*c^2)*d*e^5 + 1287*(b^3 + 6*a*b*c)*e^6)*x^3 - 3*(320*c^3*d^4*e^2 - 1040*b*c^2*d^3*e^3 + 1144*(b^2*c +
a*c^2)*d^2*e^4 - 429*(b^3 + 6*a*b*c)*d*e^5 - 6006*(a*b^2 + a^2*c)*e^6)*x^2 + (1280*c^3*d^5*e - 4160*b*c^2*d^4
*e^2 + 15015*a^2*b*e^6 + 4576*(b^2*c + a*c^2)*d^3*e^3 - 1716*(b^3 + 6*a*b*c)*d^2*e^4 + 6006*(a*b^2 + a^2*c)*d*
e^5)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 6.76612, size = 405, normalized size = 1.61 \begin{align*} \frac{2 \left (\frac{2 c^{3} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{5}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (5 b c^{2} e - 10 c^{3} d\right )}{11 e^{5}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (4 a c^{2} e^{2} + 4 b^{2} c e^{2} - 20 b c^{2} d e + 20 c^{3} d^{2}\right )}{9 e^{5}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 a^{2} c e^{4} + 2 a b^{2} e^{4} - 12 a b c d e^{3} + 12 a c^{2} d^{2} e^{2} - 2 b^{3} d e^{3} + 12 b^{2} c d^{2} e^{2} - 20 b c^{2} d^{3} e + 10 c^{3} d^{4}\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{2} b e^{5} - 2 a^{2} c d e^{4} - 2 a b^{2} d e^{4} + 6 a b c d^{2} e^{3} - 4 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} - 4 b^{2} c d^{3} e^{2} + 5 b c^{2} d^{4} e - 2 c^{3} d^{5}\right )}{3 e^{5}}\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a)**2*(e*x+d)**(1/2),x)

[Out]

2*(2*c**3*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(5*b*c**2*e - 10*c**3*d)/(11*e**5) + (d + e*x)**(9/2
)*(4*a*c**2*e**2 + 4*b**2*c*e**2 - 20*b*c**2*d*e + 20*c**3*d**2)/(9*e**5) + (d + e*x)**(7/2)*(6*a*b*c*e**3 - 1
2*a*c**2*d*e**2 + b**3*e**3 - 12*b**2*c*d*e**2 + 30*b*c**2*d**2*e - 20*c**3*d**3)/(7*e**5) + (d + e*x)**(5/2)*
(2*a**2*c*e**4 + 2*a*b**2*e**4 - 12*a*b*c*d*e**3 + 12*a*c**2*d**2*e**2 - 2*b**3*d*e**3 + 12*b**2*c*d**2*e**2 -
20*b*c**2*d**3*e + 10*c**3*d**4)/(5*e**5) + (d + e*x)**(3/2)*(a**2*b*e**5 - 2*a**2*c*d*e**4 - 2*a*b**2*d*e**4
+ 6*a*b*c*d**2*e**3 - 4*a*c**2*d**3*e**2 + b**3*d**2*e**3 - 4*b**2*c*d**3*e**2 + 5*b*c**2*d**4*e - 2*c**3*d**
5)/(3*e**5))/e

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Giac [A]  time = 1.28703, size = 574, normalized size = 2.28 \begin{align*} \frac{2}{45045} \,{\left (6006 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a b^{2} e^{\left (-1\right )} + 6006 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2} c e^{\left (-1\right )} + 429 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} b^{3} e^{\left (-2\right )} + 2574 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a b c e^{\left (-2\right )} + 572 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} b^{2} c e^{\left (-3\right )} + 572 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} a c^{2} e^{\left (-3\right )} + 65 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} b c^{2} e^{\left (-4\right )} + 10 \,{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5}\right )} c^{3} e^{\left (-5\right )} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^2*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*b^2*e^(-1) + 6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(
3/2)*d)*a^2*c*e^(-1) + 429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*b^3*e^(-2) + 2
574*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b*c*e^(-2) + 572*(35*(x*e + d)^(9/2
) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*b^2*c*e^(-3) + 572*(35*(x*e + d
)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a*c^2*e^(-3) + 65*(315*(x
*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)
^(3/2)*d^4)*b*c^2*e^(-4) + 10*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12
870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*c^3*e^(-5) + 15015*(x*e + d)^(3
/2)*a^2*b)*e^(-1)